THE THREE NUCLEON SYSTEM AT LEADING ORDER OF CHIRAL EFFECTIVE THEORY

Transcription

1 THE THREE NUCLEON SYSTEM AT LEADING ORDER OF CHIRAL EFFECTIVE THEORY Young-Ho Song(RISP, Institute for Basic Science) Collaboration with R. Lazauskas( IPHC, IN2P3-CNRS) U. van Kolck (Orsay, IPN & Arizona U.) INT, UW,Seattle

2 Outline Introduction Chiral Effective Field Theory Weinberg Power counting scheme Problems in non-perturbative renormalization Modified power counting scheme Results Two-body case(deuteron, n-p scattering) Three-body case( triton, n-d scattering) Summary/Discussion

3 Effective Field Theory QCD at low energy scale requires either Non-perturbative calculation of QCD (Lattice QCD) Or effective formulation of strong interaction for hadrons. (Nuclear) Effective Field Theory - Basic assumption: Low energy dynamics are not sensitive to the short distance detail and we only need to know finite number of LECs at desired accuracy.

4 Chiral Perturbation Theory Chiral effective field theory Low energy effective theory of QCD for hadrons Same symmetry and symmetry breaking pattern of QCD : Chiral symmetry Infinite number of operators, diagrams Require truncation scheme or power counting: Expansion in power of Q/M_hi, Q/cutoff Require Renormalization Group invariance : cutoff independence expect: inverse power of (residual) cutoff dependence In order to maintain predictive power of EFT, it is necessary to truncate the sum in such a way that the resulting cutoff dependence and truncation error can be decreased systematically as increasing order.

5 Chiral Perturbation Theory Chiral Perturbation Theory(ChPT): EFT of pions Expansion in power of All divergences(cutoff dependence) from loop diagrams can be absorbed by counter terms order by order.

6 Heavy Baryon Chiral Perturbation Theory ChPT+ nucleon: Relativistic theory requires subtle power counting scheme because of heavy nucleon mass scale. Heavy Baryon Chiral Perturbation theory: expansion in inverse power of nucleon mass. All divergences(cutoff dependence) from loop diagrams can be absorbed by counter terms order by order.

7 Weinberg power counting scheme However, two nucleon system must be treated non-perturbatively Existence of shallow deuteron bound state Large scattering length in S-wave scattering -> Sum of infinite number of diagrams Infrared enhancement from pure nucleon intermediate state diagram (reducible diagrams)

8 Weinberg power counting scheme Weinberg s suggestion: Compute potential from irreducible diagram Non-perturbative sum of reducible diagrams with kernel Naive Dimensional Analysis of (Heavy Baryon) ChPT -> perturbative Power counting for nuclear system -> Explain the potential order 2B > 3B > 4B

9 Weinberg power counting scheme

10 Chiral EFT potential Perturbative Renormalization in kernel: Divergence(cutoff dependence) from loop diagrams can be absorbed by contact counter terms order by order. Non-perturbative Renormalization of amplitude: LS equation is usually divergent and require regularization Cutoff dependence have to be removed by renormalization

11 Problem in Weinberg power counting Implicit assumption: counter terms in potential are also enough to absorb cutoff dependence from iteration of one-pion exchange diagrams. Wrong. Example: D.B.Kaplan, M.J. Savage, M.B.Wise, NPB478,629(1996) Divergence of Leading order Diagram Cannot be absorbed by Leading order counter term

12 Problem in Weinberg power counting Example: A. Nogga, R.G.E.Timmermans, U. van Kolck PRC72,054006(2005) In addition to the 1S0, 3S1 counter terms at LO, counter terms for 3P0, 3P2,3D2 are required at LO to absorb cutoff dependence.

13 Modified power counting Origin of problem: singular tensor force in OPE Nogga, Timmerman, van Kolck (PRC72(2005)054006) Weinberg s power counting is okay for singlet and repulsive tensor channel : leading order counter term can absorb cutoff dependence However, attractive tensor channels show limit cycle like cutoff dependence (3P0, 3P2, 3D2) Need to promote higher order counter terms to leading order

14 Modified power counting This work basically reproduce NTvK results in two-body and studies cutoff dependence in 3-body system. Does three-body require modification in power counting? (like pionless EFT?) LO potential All LECs are fitted to PWA93 phase shifts at T_L= 10 MeV

15 Results : 2 body scattering 10 MeV (red solid line), 50 MeV (green dashed line), 100 MeV ( blue dotted line), and 200 MeV (magenta dot dashed line)

16 Counter terms

17 Results : 2 body bound state With Counter terms

18 Results : 2 body scattering(cutoff)

19 Results : 2 body scattering(energy) PWA93 data(red solid line), cutoff values 600 MeV(green dashed line), 1 GeV( blue dotted line), 4 GeV (magenta dot dashed line) and 10 GeV(black dots).

20 Results : 2 body scattering

21 Results : 2 body scattering

22 Results : 2 body scattering

23 Results : 2 body scattering

24 Results : 2 body scattering

25 Results : 2 body scattering(regulator) Regulator dependence

26 Results : 3 body We obtain 3-body wave function by solving Faddeeve equation in configuration space Fourier transform potential from momentum space to configuration space Removed unphysical deep bound states in 3-body calculation.

27 Results : 3 body Partial wave dependence: jmax value dependence jmax = 1 (green dashed line), jmax = 2 ( blue dotted line), jmax = 3 (magenta dot dashed line),and jmax = 4 (black dots).

28 Results : 3 body regulator dependence n=2 (green dashed line), n=4regulator ( blue dotted line), n=6 regulator (magenta dot dashed line).

29 Results: 3-body (fitting) If we fit 3S1 counter term to deuteron binding energy

30 Summary and Discussion We confirmed the results of NTvK in larger cutoff range(~10 GeV) Leading order potential in attractive tensor channels requires additional counter terms to original Weinberg power counting potential and can provide cutoff independent results. Triton binding energy and n-d scattering length result shows that it does not require modification of power counting of three body force at leading order. The NLO calculation is on-going.

31 Summary and discussion Fitting cutoff dependence To have better accuracy to experimental results and insight on the power of three-body force, We need to go beyond leading order.

32 Preliminary: NLO calculation. Long and Yang, PRC86, (2012) Note: NLO and higher are strictly treated as a perturbation.

33 Preliminary: NLO calculation Only 1S0 counter term at NLO Fit counter terms at E=20 MeV Is 3-body counter term necessary at NLO? In progress.